Answer to Number 34 on Multiple Choice Test #6

Another teacher (Armand Amaranto) and I conferred and think we have the correct answer and how to show it to students - note the problem with g at the end – we probably make a mistake somewhere

Here's the solution...

Start with

a= dv/dt = g - bv

with some algebra

dv = (g - bv) dt

doing some algebra and integrating both sides of the equation...(basically I isolated any terms with v on one side and the other side with terms with t's.

dv

Integral (---------) = Integral (dt)

g - bv

Aside:

The integral of 1/x with respect to x is the natural log of x....."lnx"

The integral of 1/(1-x) with respect to x is the -"natural log of (1-x)

...."-ln(1-x)"

The integral of 1/(1-3x) with respect to x is the -1/3"natural log of (1-x)

...."-1/3{ln(1-x)}"

Back to our problem....

Integrating both sides of the equation...we get...

-1/b ln(g-bv) = t

ln (g-bv) = -bt

Aside...ln x = y is the same as x = e^y (e raised to the y power)

so our expression becomes...

g-bv = e^(-bt) (solving for v)

-bv = e^(-bt) - g

g - e^(-bt)

v = -------------

b

So that leaves only "A" as the answer. I think the answer on the sheet is wrong since they factored out the "g".

Show it to your calculus students and I think they may know how to do this by now. I am pretty sure my answer is correct. The integral shown here seems to be appearing more and more often...it is on the 1993 AP exam...the 3rd free response question. Does this mean we should start to teach this integral? This is basically a differential equation type problem. It is not something I teach the kids.